A number of evolutionary scenarios have been proposed that lead to the formation of binary systems consisting of a degenerate dwarf in orbit with a neutron star. In fact, most degenerate dwarfs in close-binary systems are probably the cores of evolved stars whose envelopes have been stripped by one or more episodes of common-envelope evolution. Once the envelope has been removed, the newly exposed core of the giant will have an effective temperature and radius that are considerably larger than the corresponding quantities in an older, highly degenerate dwarf. It is important for at least two reasons to evaluate the subsequent cooling history of such objects: (i) For detached binaries, we want to be able to infer from the current effective temperature and luminosity of the degenerate dwarf the elapsed time since the core of its progenitor giant was exposed, (ii) For semi-detached binaries, we must know the evolution of the radius of the degenerate dwarf in order to compute important relationships among the orbital period and its rate of change, the mass of the degenerate dwarf, and the mass-transfer rate.